Please answer all the parts of the following question correctly. Please use the correct notation and answer the parts in the correct order. Please show all your working/solution(s). Kindly box your final answer.

?

Consider a datagenerating process where X N Unif(0, 1) and Y = exp(X) + e where e | X ~ N(0, 1). (a) Use R to generate 500 independent observations from this process, and plot them. (b) Find lE(Y). (c) Find the true value of the slope for the best linear predictor ,81 = %%1 Hint: Use integration by parts for IE(XY), and recall that for X uniform on (0,1) we have IE(X) = % and Var(X) = . ((1) Find the true intercept for the best linear predictor g 2 lE(Y) 611E(X). (e) Use 1m in R to regress Y on X with the 500 observations you generated in (a). Are the estimated coefcients close to the true values you found in (c) and (d)? What if you do a regression with 1 million observations? (f) Add three lines to the plot from (a) based on 500 observations, each with a dierent color: (i) the estimated best regression line, (ii) the true best regression line, and (iii) the true regression function (i.e., true conditional expectation of Y given X)